feat: error handling, logging, analytics, OG tags, and sticky sidebar (#44)

## Summary

- Add comprehensive try/catch error handling across all API routes, middleware, and the Hono global error handler, piping every unhandled error to the `@nhcarrigan/logger` service to prevent silent crashes and unhandled Promise rejections
- Add a `logError` utility on the frontend that forwards errors through the overridden `console.error` to the backend telemetry endpoint; apply it to every silent `catch {}` block in the game context, sound, notification, and clipboard utilities, and wrap the React tree in an `ErrorBoundary`
- Add Plausible analytics, Open Graph + Twitter Card meta tags, Tree-Nation widget, and Google Ads to `index.html`
- Make the game sidebar sticky with a `--resource-bar-height` CSS custom property offset so it stays viewport-height without overlapping the resource bar; reset sticky behaviour in the mobile responsive override

## Test plan

- [ ] Lint passes: `pnpm lint`
- [ ] Build passes: `pnpm build`
- [ ] Verify errors thrown in API routes appear in the logger service rather than crashing the process
- [ ] Verify frontend errors appear in the `/api/fe/error` backend log
- [ ] Verify Open Graph tags render correctly when sharing the URL
- [ ] Verify Plausible analytics fires on page load
- [ ] Verify Tree-Nation badge renders in the sidebar
- [ ] Verify sidebar stays fixed while the main content scrolls on desktop
- [ ] Verify mobile layout is unaffected

✨ This issue was created with help from Hikari~ 🌸

Reviewed-on: #44
Co-authored-by: Hikari <hikari@nhcarrigan.com>
Co-committed-by: Hikari <hikari@nhcarrigan.com>

apps/web/src/components/game/characterSheetPanel.tsx

@@ -19,6 +19,7 @@ import {
 import { type ChangeEvent, type JSX, useEffect, useRef, useState } from "react";
 import { updateProfile } from "../../api/client.js";
 import { useGame } from "../../context/gameContext.js";
+import { logError } from "../../utils/logError.js";
 
 interface EquippedItem {
   name:   string;
@@ -205,12 +206,16 @@ const CharacterSheetPanel = (): JSX.Element => {
   function handleShareClick(): void {
     const discordId = player?.discordId ?? "";
     const url = `${window.location.origin}/character/${discordId}`;
-    void navigator.clipboard.writeText(url).then(() => {
-      setCopied(true);
-      setTimeout(() => {
-        setCopied(false);
-      }, 2000);
-    });
+    void navigator.clipboard.writeText(url).
+      then(() => {
+        setCopied(true);
+        setTimeout(() => {
+          setCopied(false);
+        }, 2000);
+      }).
+      catch((error_: unknown) => {
+        logError("clipboard_copy", error_);
+      });
   }
 
   function handleNameChange(event: ChangeEvent<HTMLInputElement>): void {